Problem 127 - Precalculus with Limits

Question

In Exercises 127 - 130, use the following information. A tax rebate has been given to property owners by the state government with the anticipation that each property owner will spend approximately \( p\% \) of the rebate, and in turn each recipient of this amount will spend \( p\% \) of what they receive, and so on. Economists refer to this exchange of money and its circulation within the economy as the multiplier effect. The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the states economy, if this effect continues without end. Tax rebate \( \$400 \) \( p\% \) \( 75\% \)

Step-by-Step Solution

Answer

The total amount put back into the state's economy if this effect continues without end is \$1600.

Step 1: Identify the first term and common ratio

Let's identify the parameters needed. The first term (\( a \)) is the amount of the initial tax rebate, which is \$400. The common ratio (\( r \)) is the percentage of the rebate that is being spent on each cycle. It's given as 75%, but we will use it in decimal form as 0.75.

Step 2: Apply the formula for the Sum of a geometric series

The formula for the sum of an infinite geometric series is \( S = \frac{a}{1 - r} \). Substitute \( a = \$400 \) and \( r = 0.75 \) into the formula.

Step 3: Calculate the Sum

After substituting, the formula becomes \( S = \frac{\$400}{1 - 0.75} = \frac{\$400}{0.25} \). Solving the equation gives \( S = \$1600 \).

Key Concepts

Multiplier Effect in Economics

The multiplier effect is a fascinating concept in economics that describes how spending ripples through an economy.
It captures the idea that one person's spending becomes another person's income, creating a cycle of economic activity.
For instance, when a government provides a tax rebate, individuals tend to spend a portion of it. This spending is not just a singular event. It triggers a chain reaction:

• The first recipient spends a substantial portion of the money.
• The people who receive this money then spend a part of it on their own needs.
• This cycle continues indefinitely, as each transaction becomes part of the ongoing chain.

In the context of the given exercise, the initial amount (\(\(400\)) creates an economic wave, magnifying its impact as it moves through the economy. The initial spending by the property owner is much louder in its echo due to this multiplier effect, resulting in a much larger total impact (\(\)1600\)). By understanding how spending circulates, economists can better predict and influence economic growth.

Understanding Infinite Series

An infinite series is a sequence of numbers that continue indefinitely.
These are commonly encountered in geometric series, where each term is derived from multiplying the previous one by a constant factor.
The specific form used here describes an entire cycle of economic activity continuing forever. Even though it seems never-ending, calculating an infinite series's total—like in the tax rebate scenario—uses specific formulas to get a finite result.
In the exercise, each recipient spends a percentage of their received money, setting up a geometric sequence:

• Start with an initial amount.
• Successively spend a fraction to form subsequent terms in the series.
• This forms a mathematical pattern that repeats infinitely.

This pattern can be expressed in a formal equation, allowing calculation of the total sum despite its infinite nature. The convergence here, despite infinite terms, leads to a discerning end due to its specified decay (\(75\%\) multiplier).

Unpacking the Common Ratio

The common ratio in a geometric series is a pivotal element since it dictates how the series behaves.
It represents the constant factor by which each term is multiplied to get the next term in the sequence.
In the exercise's scenario, the common ratio is essential to understanding the infinitely continuing economic cycle.
This is depicted in the given formula for the sum of an infinite geometric series: \[S = \frac{a}{1 - r}\]where:

• \(a\) is the first term of the series.
• \(r\), the common ratio, must be a number between -1 and 1 for the series to converge.

In our \(75\%\) scenario, the common ratio is \(0.75\). Each round in the cycle reduces the remaining amount significantly but gradually hones in upon a final limit. It is critical because a larger ratio close to 1 indicates that the terms reduce slowly, therefore the series will have a greater cumulative value eventually. This ratio is instrumental in calculating the expanded total value seen in the multiplier effect.